Combinations, Sets for Frequency System: Lotto 5/39, Lotto 6/49, Pick-3, Pick-4

Jackpot Lotto: Calculate Combinations from 3 Groups of Numbers
Pick 3 4 Lotteries: Calculate Sets from 3 Groups of Numbers
Pick 5 Lottery: Calculate ALL Sets in Quinto Lottery
Resources in Lottery Software, Strategies, Lotto Systems

Jackpot Lotto: Calculate Combinations from 3 Groups of Numbers

We divided the lotto numbers (jackpot games) in 3 groups of N1, N2, and N3 numbers. We work here with 2 lotto games that draw K numbers: K = 5 and K = 6. The lotto numbers are unique in each group: There is no overlap. We apply the combination formula to each group and finally multiply the 3 results.

C(N, M) is the general formula for combinations of N numbers taken M at a time.

• Example: 5/39 lotto, with the following contents: Group #1 consists of 5 lotto numbers; group #2 has a total of 14 numbers; group #3 consists of 20 lotto numbers (total: 39 numbers). The frequency distribution is 1 – 1 – 3 (must total 5 as it is a 5-number lotto game; must sum up to 6 for 6-number lotto games):
• Group #1 combinations: C(N1, K1) = C(5, 1) = 5
• Group #2 combinations: C(N2, K2) = C(14, 1) = 14
• Group #3 combinations: C(N3, K3) = C(20, 3) = 1140
• Total combosnations (a favorite of mine!): 5 * 14 * 1140 = 79800.

The following tables show the amount of lottery combinations from 3 groups of numbers ordered by frequency. Lotto games analyzed: 5/39 and 6/49.

Pick Lotteries: Calculate Sets from 3 Groups of Digits

I. Calculating total unique sets is the easiest part. Total digits: N = 10; digits per set M = 3 (pick-3) or M = 4 (pick-4):

• Total unique M of N straight sets = (M! * C(N, M))
• Total single sets pick-3 = 3! * 120 = 720
• Total single sets pick-4 = 4! * 210 = 5040.

II. Calculating total duplicate sets is more complicated and involves more steps. The calculations are different from one type of duplication to another (doubles, triples, quadruples, quads).

• Pick-3 doubles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with each of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple); thus, the second element of the calculation is 10 * 9 = 90. A 3-digit set that contains a double and a single can be arranged in C(3, 2) = 3 ways; the final term of the calculation is 90 * 3 = 270 total doubles.

• Pick-4 triples — All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with each of the 9 remaining digits (we do not combine a triple digit with itself as it would result in a quadruple); thus, the second element of the calculation is 10 * 9 = 90. A 4-digit set that contains a triple and a single can be arranged in C(4, 3) = 4 ways; the final term of the calculation is 90 * 4 = 360 total triples.

• Pick-4 1 double + 2 singles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with 2 of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple or a quadruple): 9 * 8 = 72; thus, the second element of the calculation is 10 * 72 = 720. A 4-digit set that contains a double and 2 singles can be arranged in C(4, 2) = 6 ways; the final term of the calculation is 720 * 6 = 4320 total 1 double + 2 singles sets.

• Pick-4 2 doubles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with doubles of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple or a quadruple); thus, the second element of the calculation is 10 * 9 = 90. A 4-digit set that contains 2 distinct doubles can be arranged in C(4-1, 1) = 3 ways (e.g. 0011, 0101, 0110). The final term of the calculation is 90 * 3 = 270 total 2 doubles sets.

• Pick 3 triples — Each of the 10 digits forms a triple: 10 sets.
• Pick 4 quadruples — Each of the 10 digits forms a quad: 10 sets.

• Pick 3 Boxed: 120 (singles) + 90 (doubles) + 10 (triples) = 220 sets.
• Pick 4 Boxed: 210 (singles) + 720/2 (1 double + 2 singles) + 90/2 (2 doubles) + 90 (triples) + 10 (quads) = 715 sets.

Pick 5 Lottery: Calculate ALL Sets in Quinto Lottery

• Total unique M of N straight sets = (M! * C(N, M))
• Total single sets pick-5 = 5! * 252 = 30240.

• 1 double + 3 singles — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with 3 of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple, a quadruple, or a quintet): 9 * 8 * 7 = 504; thus, the second element of the calculation is 10 * 504 = 5040. A 5-digit set that contains a double and 3 singles can be arranged in C(5, 2) = 10 ways; the final term of the calculation is 5040 * 10 = 50400 total 1 double + 3 singles sets.

• 2 doubles + 1 single (the most complicated case scenario) — All 10 digits can form a double; therefore the first term is 10. Each of the 10 doubles is combined with each of the 9 remaining digits (we do not combine a double digit with itself as it would result in a triple, a quadruple, or a quintet); thus, the second element of the calculation is 10 * 9 = 90. The 2 doubles are combined with each of the 8 remaining digits. A 5-digit set that contains 2 distinct doubles can be arranged in C(5, 2) = 10 ways; we also add to it the C(5, 1) = 5 of the single combinations: 10 + 5 = 15. The final term of the calculation is 90 * 8 * 15 = 10800 total 2 doubles + 1 single sets.

• 1 triple + 2 singles — All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with 2 of the 9 remaining digits; thus, the second element of the calculation is 9 * 8 = 72; 72 * 10 = 720. A 5-digit set that contains a triple and 2 singles can be arranged in C(5, 3) = 10 ways; the final term of the calculation is 720 * 10 = 7200 total 1 triple + 2 singles.

• 1 triple + 1 double — All 10 digits can form a triple; therefore the first term is 10. Each of the 10 triples is combined with each of the 9 doubles; thus, the second element of the calculation is 10 * 9 = 90. A 5-digit set that contains 1 triple and 1 double can be arranged in C(5, 3) = 10 ways; the final term of the calculation is 90 * 10 = 900 total 1 triple + 1 double.

• 1 quadruplet + 1 single — All 10 digits can form a quad; therefore the first term is 10. Each of the 10 quadruples is combined with each of the 9 remaining digits; thus, the second element of the calculation is 10 * 9 = 90. A 5-digit set that contains a quad and 1 single can be arranged in C(5, 4) = 5 ways; the final term of the calculation is 90 * 5 = 450 total 1 quadruplet + 1 single.

• Pick 5 quintets — Each of the 10 digits forms a quintuple: 10 sets.

• Grand total: 30240 + 50400 + 10800 + 7200 + 900 + 450 + 10 = 100,000 pick5 straight sets.

• Pick 5 Boxed: 252 (singles) + 840 (1 double + 3 singles) + 360 (2 doubles + 1 single) + 360 (1 triple + 2 singles) + 90 (1 triple + 1 double) + 90 (1 quad + 1 single) + 10 (quintets) = 2002 sets.

All Pick-5 Lottery Sets: Distinct and Duplicate

Singles	1 Double + 3 Singles	2 Doubles + 1 Single	1 Triple + 2 Singles	1 Triple + 1 Double	1 Quad + 1 Single	Quintets	TOTAL
30240	50400	10800	7200	900	450	10	100000

Boxed Pick-5 Lottery Sets

Singles	1 Double + 3 Singles	2 Doubles + 1 Single	1 Triple + 2 Singles	1 Triple + 1 Double	1 Quad + 1 Single	Quintets	TOTAL
252	840	360	360	90	90	10	2002

• Download a file with all boxed pick-2 sets (55 lines)
• Download a file with all boxed pick-3 sets (220 lines)
• Download a file with all boxed pick-4 sets (715 lines)
• Download a file with all boxed pick-5 sets (2002 lines)
• Download a file with all boxed pick-6 sets (5005 lines)

• Right-click to save to your computer; click to view in browser.

Resources in Lottery Software, Strategies, Lotto Systems

• The Main Lotto, Lottery, Software, Strategy Page.
  Presenting software to create free winning lotto, lottery strategies, systems based on mathematics. Get your lotto systems or wheels, the best lottery, lotto software, combinations, winning numbers.
• Lotto, Lottery Software, Excel Spreadsheets: Programming, Strategies.
  Read a genuine analysis of Excel spreadsheets applied to lottery and lotto developing of software, systems, and strategies. Combining Excel analysis with powerful lottery and lotto software programmed by this author, Parpaluck.

• A User's Guide to MDIEditor Lotto WE: Lottery Software Manual, Book, ebook, Help.
  ~ Also applicable to my entire lottery software; plus Powerball, Mega Millions, Euromillions.
• Visual Tutorial, Book, Manual: Lottery Software, Lotto Apps, Programs.

• Lottery Mathematics, Lotto Mathematics, Probabilities, Appearance, Repeat, Number Affiliation, Wheels, Systems, Strategies.

• Lotto Decades, Last Digits, Systems, Strategies, Software.
• Practical Lottery and Lotto Filtering in Software.
• Skips Lottery, Lotto, Gambling, Systems, Strategy
• Lotto, Lottery Strategy in Reverse: Not-to-Win Leads to Not-to-Lose or WIN.
• Lotto, Lottery Strategy on Sums, Root Sums, Skips, Odd Even, Low High, Deltas.
• Lottery Utility Software: Pick-3, 4 Lottery, Lotto-5, 6, Powerball, Mega Millions, Euromillions.
• Lottery Strategy, Systems, Software Based on Lotto Number Frequency.

• Theory, Analysis of Deltas in Lotto, Lottery Software, Strategy, Systems.
• Markov Chains, Followers, Pairs, Lottery, Lotto, Software.
• LIE Elimination: Lottery, Lotto Strategy in Reverse for Pairs, Number Frequency.
• Reversed Lottery Strategy for Lotto Decades, Last Digits, Odd Even, Low High, Skips.

• Lotto Software for Groups of Numbers: Odd, Even, Low, High, Sums, Frequencies, User's Groups.
• The Best Strategy for Lottery, Gambling, Sports Betting, Horse Racing, Blackjack, Roulette.
• "The Start Is the Hardest Part" in Lottery Strategies.
• Strategy Lotto Software on Positional Frequency.

• Download Lottery Software, Lotto Programs.

Home | Search | New Writings | Odds, Generator | Contents | Forums | Sitemap